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Elastic beams and columns
Abstract
In this chapter we consider bars (beams and columns), where one direction is substantially bigger than the other two. First in-plane problems are considered including the shear deformations. The shortcomings of the applied theories are emphasized. Then both Saint-Venant and restrained warping induced torque are presented and their role is discussed.
In the design of composite sections, beam theories are used which require the knowledge of the cross-sectional properties, that is, the bending-, the shear-, the torsional-, warping-, axial stiffnesses and the coupling terms. In the classical analysis, the properties are calculated by assuming kinematical relationships (e.g. cross sections remain plane after the deformation of the beam). These assumptions may lead to inaccurate or contradictory results. In this paper, a new theory is presented in which no kinematical assumption is applied, rather the properties are derived from the accurate (three dimensional) equations of beams using limit transition. The theory includes both the in-plane and the torsional-warping shear deformations. As a result of the analysis, the stiffness matrix of the beam is obtained which is needed for either analytical or numerical finite element (FE) solutions. Applications for open section and closed section beams are also presented.